Logics of Finite Hankel Rank
(For Yuri Gurevich at his 75th Birthday)
Abstract
We discuss the FefermanVaught Theorem in the setting of abstract model theory for finite structures. We look at sumlike and productlike binary operations on finite structures and their Hankel matrices. We show the connection between Hankel matrices and the FefermanVaught Theorem. The largest logic known to satisfy a FefermanVaught Theorem for productlike operations is , first order logic with modular counting quantifiers. For sumlike operations it is , the corresponding monadic second order logic. We discuss whether there are maximal logics satisfying FefermanVaught Theorems for finite structures.
1 Introduction
1.1 Yuri’s Quest for Logics for Computer Science
The second author (JAM) first met Yuri Gurevich in spring 1976, while being a Lady Davis fellow at the Hebrew University, on leave from the Free University, Berlin. Yuri had just recently emigrated to Israel. Yuri was puzzled by the supposed leftist views of JAM, perceiving them as antagonizing. This lead to heated political discussions. In the following time, JAM spent more visiting periods in Israel, culminating in the Logic Year of 1980/81 at the Einstein Institute of the Hebrew University, after which he finally joined the Computer Science Department at the Technion in Haifa.
At this time both Yuri and JAM worked on chapters to be published in [1], Yuri on Monadic Second Order Logic, and JAM on abstract model theory. Abstract model theory deals with metamathematical characterizations of logic. Pioneered by P. Lindström, G. Kreisel and J. Barwise, in [1, 2],[26],[30, 31], First Order Logic and admissible fragments of infinitary logic were characterized. Inspired by H. Scholz’s problem, [10], R. Fagin initiated similar characterizations when models are restricted to finite models, connecting finite model theory to complexity theory.
At about the same time Yuri and JAM both underwent a transition in research orientation, slowly refocusing on questions in theoretical Computer Science. Two papers document their evolving views at the time, [23],[37]. Yuri was vividly interested in [37] and frequent discussions between Yuri and JAM between 1980 and 1982 shaped both papers. In [37] the use for theoretical computer science of classical model theoretic methods, in particular, the role of the classical preservation theorems (see below), was explored, see also [34],[36]. Yuri grasped early on that these preservation theorems do not hold when one restricts First Order Logic to finite models.
Under the influence of JAM’s work in abstract model theory, the foundations of database theory and logic programming, [5, 6],[34],[36],[38],[43], and the work of N. Immerman and M. Vardi, [24],[47], Yuri stressed the difference between classical model theory and finite model theory. In [23], he formulated what he calls the Fundamental Problem of finite model theory. This problem is, even after 30 years, still open ([23]): Is there a logic such that any class of finite structures is definable in iff is recognizable in polynomial time. For ordered finite structures there are several such logics, [21],[24],[39],[41, 42],[47]. We give a precise statement of the Fundamental Problem in section 2, Problem 1.
1.2 Preservation Theorems
Let be two syntactically defined fragments of a logic , and let be a binary relation between structures. Preservation theorems are of the form:
Let . The following are equivalent:
For all structures with , we have that if satisfies , then also satisfies .
There is which is logically equivalent to .
A typical example is Tarski’s Theorem for first order logic, with all of first order logic, its universal formulas, and holds if is a substructure of . Many other preservation theorems can be found in [7]. In response to [37],[43], Yuri pointed out in [23] that most of the preservation theorems for first order logic fail when one restricts models to be finite.
1.3 Reduction Theorems
Let denote the finite sequences of formulas in , and let be a binary operation on finite structures. Reduction theorems are of the form:
There is a function with and a Boolean function such that for all structures and all , the structure satisfies iff
(1)
where for we have iff and iff . There are also versions for ary operations .
The most famous examples of such reduction theorems are the FefermanVaughttype theorems, [12, 13, 14, 15],[22],[40]. A simple case is Monadic Second Order Logic (), where and is the disjoint union of and . Additionally it is required that the quantifier ranks of the formulas in do not exceed the quantifier rank of . In [38, Chapter 4] such reduction theorems are discussed in the context of abstract model theory. However, in [38, Chapter 4] the quantifier rank has no role.
In contrast to preservation theorems, reduction theorems still hold when restricted to finite structures.
1.4 Purpose of this Paper
In [40] JAM discussed FefermanVaughttype theorems in finite model theory and their algorithmic uses. In Section 7 of that paper, it was asked whether one can characterize logics over finite structures which satisfy the FefermanVaught Theorem for the disjoint union . The purpose of this paper is to outline new directions to attack this problem. The novelty in our approach is in relating the FefermanVaught Theorem to Hankel matrices of sumlike and connectionlike operations on finite structures. Hankel matrices for connectionlike operations, aka connection matrices, have many algorithmic applications, cf. [28],[33].
In section 2 we set up the necessary background on Lindström logics, quantifier rank, translation schemes, and sumlike operations. A Hankel matrix involves a binary operation on finite structures which results in a structure, and a class of structures closed under isomorphisms (aka a property). In section 3 we give the necessary definitions of Hankel matrices and their rank. We then study properties where has finite rank. We show that there are uncountably many such properties and state that the class of all properties that have finite rank for every sumlike operation forms a Lindström logic, Theorems 2 and 5. In section 4 we define various forms of FefermanVaughttype properties of Lindström logics equipped with a quantifier rank, and discuss their connection to Hankel matrices. Theorem 11 describes their exact relationship. A logic has finite Srank, if all its definable properties have Hankel matrices of finite rank for every sumlike operation. In section 5 we sketch how to construct a logic satisfying the FefermanVaught Theorem for sumlike operations from a logic which has finite Srank. Finally, in section 6, we discuss our conclusions and state open problems. A full version of this paper is in preparation, [27].
2 Background
2.1 Logics with Quantifier Rank
We assume the reader is familiar with the basic definitions of generalized logics, see [1],[11]. We denote by finite relational vocabularies, possibly with constant symbols for named elements. structures are always finite unless otherwise stated. A finite structure of size is always assumed to have as its universe the set . A class of finite structures closed under isomorphisms is called a property.
A Lindström Logic is a triple
where is the set of sentences of , are the finite structures, is the satisfaction relation. The satisfaction relation is a ternary relation between structures, assignments and formulas. An assignment for variables in a structure is a function which assigns to each variable an element of the universe of . We always assume that the logic contains all the atomic formulas with free variables, and is closed under Boolean operations and first order quantifications. A logic is a sublogic of a logic iff for all and the satisfaction relation of is the satisfaction relation induced by .
A Gurevich logic is a Lindström logic where additionally the sets are uniformly computable.
A Lindström logic with a quantifier rank is a quadruple
where additionally is a quantifier rank function. A quantifier rank (qrank) is a function such that

For atomic formulas the qrank .

Boolean operations and translations induced by translation schemes (see subsection 2.2) with formulas of qrank preserve maximal qrank.
A quantifier rank is nice if additionally it satisfies the following:

For finite , there are, up to logical equivalence, only finitely many formulas of fixed qrank with a fixed set of free variables.
In the presence of (iii) we define Hintikka formulas as maximally consistent formulas of fixed qrank. A nice logic is Lindström logic with a nice quantifier rank . We note that in a nice logic, the only formulas of qrank are Boolean combinations of atomic formulas.
We denote by , , , first order, monadic second order, and full second order logic, respectively. All these logics are nice Gurevich logics with their natural quantifier rank, and they are sublogics of .
We denote by , , first order and monadic second order logic augmented by the modular counting quantifiers which say that there are modulo , exactly many elements satisfying . In the presence of the quantifier there are two definitions of the quantifier rank: and . Given any finite set of variables, for we have, up to logical equivalence, infinitely many formulas with , whereas for there are only finitely many such formulas. and with the quantifier rank are nice Gurevich logics. In the sequel we always use as the quantifier rank for and .
, fixed point logic, is also a Gurevich logic and a sublogic of . However, order invariant is a sublogic of which is not a Lindström logic. The definable properties in order invariant are exactly the properties recognizable in polynomial time. For and order invariant see [21],[24],[39],[41, 42],[47].
Problem 1 (Y. Gurevich, [23]).
Is there a Gurevich logic such that the definable properties are exactly the properties recognizable in polynomial time.
2.2 SumLike and ProductLike Operations on
structures
The following definitions are taken from [40]. Let be two relational vocabularies with , and denote by the arity of . A translation scheme is a sequence of formulas where has free variables, and each has free variables. In this paper we do not allow redefining equality, nor do we allow name changing of constants.
We associate with two mappings and , the transduction and translation induced by . The transduction of a structure is the structure where the vocabulary is interpreted by the formulas given in the translation scheme. The translation of a formula is obtained by substituting atomic formulas with their definition through formulas given by the translation scheme. A translation scheme (induced transduction, induced translation) is scalar if , otherwise it is vectorized. It is quantifierfree if so are the formulas .
If has no constant symbols, the disjoint union of two structures is the structure obtained by taking the disjoint union of the universes and of the corresponding relation interpretations in and . On the other hand, if has finitely many constant symbols the disjoint union of two structures is a structure with twice as many constant symbols, . Connection operations are similar to disjoint unions with constants, where equally named elements are identified. We call the disjoint union followed by the pairwise identification of constant pairs the sum, cf. [33].
A binary operation is sumlike (productlike) if it is obtained from the disjoint union of structures by applying a quantifierfree scalar (vectorized) transduction. A binary operation is connectionlike if it is obtained from a connection operation on structures by applying a quantifierfree scalar transduction. If , we say is an operation on structures.
Connectionlike operations are not sumlike according to the definitions in this paper^{1}^{1}1They are nevertheless called sumlike in [40].. Although connection operations are frequently used in the literature, cf. [33],[40], we do not deal with them in this paper. Most of our results here can be carried over to connectionlike operations, but the formalism required to deal with the identification of constants is tedious and needs more place than available here.
Proposition 0.
Let be a fixed finite relational vocabulary.

There are only finitely many sumlike binary operations on structures.

There is a function such that for each there are only many vectorized productlike binary operations on structures.
2.3 Abstract Lindström Logics
In [31] a syntaxfree definition of a logic is given. An abstract Lindström logic consists of a family of properties closed under certain operations between properties of possibly different vocabularies. One thinks of as the family of definable properties. We do not need all the details here, the reader may consult [1],[30, 31]. The main point we need is that every abstract Lindström logic can be given a canonical syntax using generalized quantifiers.
3 Hankel matrices of properties
3.1 Hankel Matrices
In linear algebra, a Hankel matrix, named after Hermann Hankel, is a real or complex square matrix with constant skewdiagonals. In automata theory, a Hankel matrix is an infinite matrix where the rows and columns are labeled with words over a fixed alphabet , and the entry is given by . Here is a realvalued word function and denotes concatenation. A classical result of G.W. Carlyle and A. Paz [4] in automata theory characterizes realvalued word functions recognizable by weighted (aka multiplicity) automata in algebraic terms.
Hankel matrices for graph parameters (aka connection matrices) were introduced by L. Lovász [32] and used in [18],[33] to study realvalued partition functions of graphs. In [18],[33] the role of concatenation is played by connections of graphs, i.e., graphs with distinguished vertices .
In this paper we study matrices which are Hankel matrices of properties of general relational structures and the role of connections is played by more general binary operations, the sumlike and productlike operations introduced in [44] and further studied in [40].
Definition 1.
Let be a binary operation on finite structures returning a structure, and let be a property.

The Boolean Hankel matrix is the infinite matrix where the rows and columns are labeled by all the finite structures, and iff .

The rank of over is denoted by , and is referred to as the Boolean rank.

We say that has finite rank iff is finite.

Two structures are equivalent, , if for all finite structures we have
(2) 
For a structure , we denote by the equivalence class of .

We say that has finite index^{2}^{2}2 K. Compton and I. Gessel, [8],[19], already considered properties of finite index for the disjoint union of structures. In [17] this is called Gessel index. C. Blatter and E. Specker, in [3],[46], consider a substitution operation on pointed structures, , where the structure is inserted into at a point . is sumlike, and the index is called in [17] Specker index. iff there are only finitely many equivalence classes.
Proposition 1.
Let be a property.
has finite rank iff
has finite index.
Sketch of proof.
We first note that two structures are in the same equivalence class of iff they have identical rows in . As the rank is over , finite rank implies there are only finitely many different rows in . The converse is obvious. ∎
3.2 Properties of Finite rank
We next show that there are uncountably many properties of finite rank. We also study the relationship between the rank and rank of properties for different operations and .
We first need a lemma.
Lemma 1.
Let and let be the infinite matrix whose columns and rows are labeled by the natural numbers , and iff . Then has finite rank over iff is ultimately periodic.
Theorem 2.
Let be the vocabulary with one binary edgerelation, and be augmented by one vertex label. Let and be the graph properties defined by , , and , where is the complement graph of the clique of size , and is a path graph of size .

has finite rank for all .

For two graphs , let be the sumlike operation defined as the loopless complement graph of .
has infinite rank for all which are not ultimately periodic.
Equivalently, for the property , the Hankel matrix has infinite rank for all which are not ultimately periodic. 
has finite rank for all sumlike operations on structures and all .

For two graphs with one vertex label, i.e. structures, let be the sumlike operation defined as the graph resulting from by adding an edge between the two labeled vertices and then removing the labels. has infinite rank for all which are not ultimately periodic.

has finite rank for all .
Theorem 2 needs an interpretation: (i) says that there is a specific sumlike operation such that there uncountably many classes of structures with finite rank^{3}^{3}3 A similar construction was first suggested by E. Specker in conversations with the second author in 2000, cf. [40, Section 7].. (ii) says that if a class has finite rank for one sumlike operation, it does not have to hold for all sumlike operations^{4}^{4}4 This observation was suggested by T. Kotek in conversations with the second author in summer 2014.. (iii) produces uncountably many classes of structures which have finite rank for all sumlike operations on structures. (iv) finally shows that such classes can still have infinite rank for sumlike operations which take as inputs structures (labeled paths) and output a structure (unlabeled paths). This leads us to the following definition:
Definition 3.
Let be a vocabulary and be a property.

has finite Srank (Prank, Crank) if for every sumlike (productlike, connectionlike) operation the Boolean rank of is finite.

A nice logic has finite Srank (Prank, Crank) iff all its definable properties have finite Srank (Prank, Crank).
3.3 Proof of Theorem 2
Proof.
(i) The disjoint union of two graphs is never connected. Therefore all the entries of are zero,
unless we consider the empty graph to be structure. In this case we have exactly one row and one column representing .
In any case, the rank is .
(ii) Consider the submatrix of consisting of rows and columns labeled with the
edgeless graphs and use subsection 3.2.
(iii)
We first observe that
(*) for any sumlike operation on structures (i.e., graphs), and , if for , either or must be the empty graph.
This is due to the fact that has no constant symbols.
Therefore, a row or column containing nonzero entries must be labeled by the empty graph.
(iv)
Here we consider translation schemes for sumlike operations, with .
Hence (*) from the proof of (iii) is not true anymore because now can be obtained from
and with the being an end vertex, using .
So we apply subsection 3.2.
(v) Connection operations of two large enough cliques still produce connected graphs, but never form a clique.
∎
3.4 Properties of Finite Srank and Finite Prank
Let and denote the collection of all properties of finite Srank and finite Prank respectively, and let and .
Theorem 5.
and and are abstract Lindström logics which have finite Srank and finite Prank, respectively.
Sketch of proof:.
It is unclear whether the abstract Lindström logic goes beyond . As of now, we were unable to find a property which has finite Srank, but is not definable in .
Problem 2.

Is every property with finite Srank definable in ?

Is every property with finite Prank definable in ?
It seems to us that the same can be shown for connectionlike operations, but we have not yet checked the details.
4 Hankel matrices and the FefermanVaught theorem
4.1 The FVproperty
In this section we look at nice Lindström logics with a fixed quantifier rank. We use it to derive from the classical FefermanVaught theorem an abstract version involving the quantifier rank. This differs from the treatment in [1, Chapter xviii]. Our purpose is to investigate the connection between Hankel matrices of finite rank and the FefermanVaught Theorem on finite structures in an abstract setting.
Definition 6.
Let be a nice logic with quantifier rank .

We denote by the set of sentences (without free variables) with .

Two structures are equivalent, , if for every we have iff .

has the FVproperty for with respect to if for every there are , and such that for all structures we have that
iff
where for we have iff and iff .

is smooth with respect to if for every two pairs of structures with and we also have .
If is clear from the context we omit it.
A close inspection of the classical proofs shows that the requirements concerning the quantifier rank are satisfied in the following cases.
Examples 7.
4.2 The FVproperty and Finite Rank
Definition 8.
Let be a nice logic.

Let be a binary operation on structures. is closed if all the equivalence classes of are definable in .

is Sclosed (Pclosed, Cclosed) if for every sumlike (productlike, connectionlike) binary operation the logic is closed.
Proposition 8.
Let have the FVproperty for .

is smooth.

Let be a property definable by a formula . Then each equivalence class of is definable by a formula .

If has the FVproperty for all sumlike (productlike) operations then is Sclosed (Pclosed).
Sketch of proof.
(i) Follows because for , the truth value of depends only on , the Boolean function associated with the FVproperty.
(ii) Fix a structure . We want to show that is definable by some formula .
iff for all ,
iff
.
We have, using , that
iff for all ,
(3) 
iff ,
(4) 
where , and are as in Definition 6(iii). Equation (4) can be expressed by a formula .
(iii) Follows from (ii). ∎
By analyzing the proof in [20], one can prove:
Theorem 9.
Let be a nice Lindström logic with quantifier rank and be a binary operation on structures. If is smooth with respect to , then every definable property has finite rank.
Sketch of proof.
Let be definable by with quantifier rank .
Now let be an enumeration of maximally consistent sentences (aka Hintikka sentences).
By our assumption is nice, so this is a finite set.
Furthermore is logically equivalent to a disjunction with ,
any every structure satisfies exactly one .
Now we use the smoothness of .
If are two structures satisfying the same , then their rows (columns) in are identical.
Hence the rank of is at most , or when empty structures are allowed.
∎
Combining Theorem 9 with subsection 4.2(i) we get:
Corollary 10.
Let be a nice Lindström logic which has the FVproperty for the binary operation , and let be definable in . Then is finite.
Proposition 10.
Let be a nice logic with quantifier rank and be a fixed operation on structure, which is associative. Assume further that for every ,

the rank of is finite, and

all equivalence classes of are definable with formulas of with quantifier rank .
Then has the FVproperty for .
We have now shown that having the FVproperty for implies that is smooth, and that smoothness implies finite rank, or equivalently, finite index.
In fact we have:
Theorem 11.
Let be a nice Sclosed logic and let be a sumlike operation. Then the following are equivalent:

has the FVproperty for every sumlike operation .

is smooth.

For all and every sumlike , the rank of is finite.

For all and every sumlike , the index of is finite.
The same holds if we replace Sclosed and sumlike by Pclosed and productlike.
Proof.
(i) implies (ii) is subsection 4.2.
(ii) implies (iii) is Theorem 9.
(iii) is equivalent to (iv) by subsection 3.1.
Finally, (iii) implies (i) is subsection 4.2.
∎
5 The Sclosure of a nice logic
Let be a nice logic of finite Srank with quantifier rank . We define to be the smallest Lindström logic such that for all sumlike
and all definable properties , all the equivalence classes of are also definable in . This gives us a Lindström logic which is Sclosed. However, in order to be a nice logic, we have to extend to in such a way that ensures it is still nice.
We proceed inductively. Recall that there are only finitely many sumlike operations for fixed and . Let where is a relation symbol of arity or a constant symbol of arity . Two vocabularies are similar if they have the same number of symbols of the same arity. The effect of a sumlike operation only depends on the similarity type of and . Hence for fixed and , there are only finitely many sumlike operations.
A typical step in the induction is as follows.
Given and and a sumlike , there are only finitely many equivalence classes of . Let with be a list of these equivalence classes.
We form with quantifier rank as follows: If is not definable in then we add it to using a Lindström quantifier with quantifier rank .
is a Lindström logic. We have to show that is nice, i.e., for fixed and fixed number of free variables, is finite up to logical equivalence. This follows from the fact that we only added finitely many Lindström quantifiers and that for all we have that .
For our induction we start with . is obtained by doing the typical step for each and each sumlike . is the union of all quantifier rank functions of the previous steps. We still have iterate this process by defining and and take the limit.
We finally get:
Theorem 12.
Let be nice with quantifier rank and of finite Srank. Then with quantifier rank is nice and has the FVproperty for all sumlike operations.
The details will be published in [27].
6 Conclusions and open problems
At the beginning of this paper we asked whether one can characterize logics over finite structures which satisfy the FefermanVaught Theorem for the disjoint union, or more generally, for sumlike and productlike operations on structures. The purpose of this paper was to investigate new directions to attack this problem, specifically by relating the FefermanVaught Theorem to Hankel matrices of finite rank. Theorem 11 describes their exact relationship.
We also investigated under which conditions one can construct logics satisfying the FefermanVaught Theorem. Theorem 2 shows that there are uncountably many properties which have finite rank Hankel matrices for specific sumlike operations. Theorem 5 shows the existence of maximal Lindström logics and where all their definable properties have finite rank for all sumlike, respectively productlike, operations. However, we have no explicit description of these maximal logics.
Problem 3.

Is every property with finite Prank (or both finite Prank and finite Crank) definable in ?

Is every property with finite Srank (finite Crank) definable in ?
In case the answers to the above are negative, we can ask:
Problem 4.

How many properties are there with finite Srank (Prank, Crank)?

Is there a nice Gurevich logic where all the properties in are definable?
In [40, Section 7, Conjecture 2] it is conjectured that there are continuum many nice Gurevich logics with the FVproperty for the disjoint union. Adding or from Theorem 2 for fixed as Lindström quantifiers to together with all the equivalence classes of or gives us a nice Lindström logic. However, the definable property that the complement of a graph is in has infinite rank, see Theorem 2(ii).
Problem 5.
How many different nice Gurevich logics with the FVproperty for the disjoint union are there?
A similar analysis for connectionlike operations will be developed in [27].
Acknowledgments
We would like to thank T. Kotek for letting us use his example, and for valuable discussions.
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